The proposed method can be seen as an extension of positional encoding methods [1]}, such as Fourier features [2]}, to arbitrary geometries. Furthermore, combining the Laplacian eigenfunctions in \(\mathbb {R}^N\) , we can recover a series of cosine functions when using Neumann boundary conditions and sine functions when using Dirichlet boundary conditions, which correspond to the encodings proposed in [1]}. These techniques have significantly improved the performance of neural networks, including PINNs [4]}. Nonetheless, we observed in our experiments that only using the eigenfunctions from the Laplacian with Neumann boundary conditions was enough to obtain accurate results. We see that including the physics of the problem greatly improves the results. Furthermore, we tested our method in a pure regression tasks in  and compared it to Gaussian process regression which can be naturally extended to work on manifolds. We see that neural networks with the eigenfunctions as input tend to overfit the data and the results depend heavily on the number of eigenfunctions used. This confirms the regularization effect of the physics, explaining the excellent performance in problems shown here. We also note that Gaussian process are an effective tool for regression on manifolds, but extending then to include physical knowledge of non-linear problems is cumbersome, requires approximate inference techniques and will suffer from the classical poor scaling of Gaussian process [5]}.
